# Systematic Design of Phononic Band Gap Crystals for Elastic Waves at the Specified Target Frequency via Topology Optimization

## TL;DR

This paper introduces a new method to design phononic crystals that can block elastic waves at specific target frequencies.

## Contribution

A topology optimization strategy using MFSE and Kriging is proposed to design phononic band gaps at specified frequencies.

## Key findings

- The method successfully optimizes band gaps for out-of-plane, in-plane, and complete wave modes.
- The approach is effective for dual-target frequency scenarios.
- Numerical results confirm the method's ability to meet bespoke frequency specifications.

## Abstract

Phononic band gap crystals are characterized by periodic scatterers embedded within a matrix, which enable precise modulation of acoustic or elastic waves. Conventional optimization prioritizes bandwidth maximization, yet practical engineering often requires band gaps at specified frequencies. This requirement creates a significant design challenge. To this end, we develop a topology optimization strategy capable of maximizing elastic wave band gaps around prescribed target frequencies. The approach utilizes Material-Field Series Expansion (MFSE) for unit cell representation and a gradient-free Kriging-based algorithm to tackle the complex optimization problems. This strategy is systematically applied to optimize the band gaps of out-of-plane, in-plane, and complete wave modes, and is further extended to more complex scenarios involving dual-target frequencies. A variety of numerical results demonstrate the method’s effectiveness in engineering phononic crystals for bespoke frequency specifications.